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A225598
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Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.
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2
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1, 1, 3, 1, 3, 5, 1, 5, 5, 9, 1, 5, 8, 9, 12, 1, 7, 11, 15, 12, 20, 1, 7, 14, 19, 19, 20, 25, 1, 9, 17, 29, 24, 33, 25, 38, 1, 9, 23, 33, 36, 42, 39, 38, 49, 1, 11, 26, 47, 46, 61, 49, 61, 49, 69, 1, 11, 32, 55, 63, 76, 70, 76, 76, 69, 87, 1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123
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OFFSET
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1,3
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COMMENTS
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For the definition of region see A206437.
T(n,k) is also the sum of all parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).
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LINKS
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EXAMPLE
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For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the sum of all parts is 3 + 1 + 1 + 3 = 8, so T(5,3) = 8.
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. Diagram Illustration of parts ending in column k:
. for n=5 k=1 k=2 k=3 k=4 k=5
. _ _ _ _ _ _ _ _ _ _
. |_ _ _ | _ _ _ |_ _ _ _ _|
. |_ _ _|_ | |_ _ _| _ _ _ _ |_ _|
. |_ _ | | _ _ |_ _ _ _| |_|
. |_ _|_ | | |_ _| _ _ _ |_ _| |_|
. |_ _ | | | _ _ |_ _ _| |_| |_|
. |_ | | | | _ |_ _| |_| |_| |_|
. |_|_|_|_|_| |_| |_| |_| |_| |_|
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k = 1 2 3 4 5
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The 5th row lists: 1 5 8 9 12
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Triangle begins:
1;
1, 3;
1, 3, 5;
1, 5, 5, 9;
1, 5, 8, 9, 12;
1, 7, 11, 15, 12, 20;
1, 7, 14, 19, 19, 20, 25;
1, 9, 17, 29, 24, 33, 25, 38;
1, 9, 23, 33, 36, 42, 39, 38, 49;
1, 11, 26, 47, 46, 61, 49, 61, 49, 69;
1, 11, 32, 55, 63, 76, 70, 76, 76, 69, 87;
1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123;
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CROSSREFS
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Cf. A000041, A066186, A135010, A141285, A186114, A186412, A187219, A194446, A206437, A207779, A211978, A225597, A225600, A225610.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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